Mathematics as a Second Language

Transcript Mathematics as a Second Language

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Arithmetic Revisited

Prelude to Mathematics as a Second Language

There is a “tongue-in-cheek” piece of advice to the effect that “ if someone doesn’t understand your explanation of something, just explain it again; only louder ”. Many students have experienced this sort of explanation too many times in their study of arithmetic; that is, listening to the same explanation being given over and over again.

To break this chain of events, we have embarked on what we believe is a fresh and productive approach.

More specifically, our innovative approach to teaching basic mathematics, which we call “ Mathematics as a Second Language ”, is to introduce numbers in the same way that people from all walks of life use them; namely as adjectives that modify nouns .

Our technique will be to show that by using this concept, all of arithmetic can be done by knowing only the addition and multiplication tables from 0 through 9. To introduce our approach, consider the following hypothetical situation.

Suppose you are the principal of an elementary school and a mother, claiming to have a precocious 5 year old son, asks to have the boy placed in a fourth grade mathematics class. You are skeptical and decide to give the youngster a quick quiz. You say to him, “Son, how much is 3 + 2?” and the boy replies “3

what

and 2

what

?” Would you now discount the mother’s claim or would you place him in the fourth grade?

Our point is that the boy’s question is very important.

Consider, for example , the true statement that… 3 dimes + 2 nickels

40 cents In this case, 3 is an adjective “dimes”, 2 is an adjective modifying modifying “nickels”, and 40 is an adjective “cents”. modifying If we omit the nouns , the above equality becomes… 3 + 2

This leads to an important result, which in this course is called the… + Fundamental Principle for Addition +

3 + 2 = 5 only when 3, 2, and 5 are adjectives that modify the same noun .

More generally, the traditional addition tables assume that the numbers being added modify the same noun .

This simple principle gives us many pieces of useful information. Namely, as trivial as it may sound, we now know that… 3 apples + 2 apples = 5 apples + = 3 people + 2 people = 5 people + = 3 cookies + 2 cookies = 5 cookies + = © 2010 Herb I. Gross

Note

The Fundamental Principle of Addition applies even if we don’t know what the noun means.

For example … 3 gloogs + 2 gloogs = 5 gloogs That is, no “them” and 2 more of “them” are 5 of “them”!

While this is not an algebra course, it is important to see how the adjective / noun theme can be incorporated into an algebra course in a natural and non-threatening way.

For example… Practice Problem #1 How much is 3x + 2x?

next Solution to the Practice Problem #1 This is a more real-world interpretation of the 3 gloogs + 2 gloogs = 5 gloogs problem. Namely, in algebra , x is used as a generic name of a particular quantity. Thus, 3 of “them” plus 2 more of “them” must be 5 of “them”. That is: 3x + 2x = 5x.

Notes on Practice Problem #1 Obviously the numerical value of 3x + 2x depends on the numerical value of x, but the fact that 3x + 2x = 5x doesn’t.

The point is that too often when students are asked to find the sum of 3x and 2x, they often act as if x is just another “gloog word” and say things like, “I don’t know what x is”.

Notes on Practice Problem #1 A more concrete illustration is to think of x as describing a colored poker chip. For example, we do not have to know how much a red chip is worth in order to know that the value of 3 red chips and 2 red chips is equal to the value of 5 red chips.

Notes on Practice Problem #1 Or on a report card, if we got five A’s, we do not usually say we got 3 A’s plus 2 A’s. On the other hand, if we got 3 A’s and 2 B’s we do not say that we got five “AB” ’s. In a similar way, in algebra if A and B are names for different nouns , we cannot add the 3 and 2 in the expression 3A + 2B.

As a classroom illustration, suppose a first or second grade student was confronted with a problem such as… 3,000,000,000 + 2,000,000,000 Because each number is represented by a ten digit numeral, this problem would be overwhelming to such a young student. © 2010 Herb I. Gross

3,000,000,000 3 billion 2,000,000,000 2 billion However, suppose we recognize that in place value notation the noun “billion” is represented by nine zeroes. That is, we may translate this problem into “plain English” by rewriting 3,000,000,000 as 3 billion and 2,000,000,000 as 2 billion © 2010 Herb I. Gross

next In this way, the relatively complicated problem 3,000,000,000 + 2,000,000,000 can be replaced by the simpler but equivalent problem 3 billion + 2 billion; the answer to which is 5 billion because 3, 2, and 5 are modifying the same noun (even though in the first or second grade the word “billion” might mean no more to a student than the word “gloog” would).

Key Point

The person who knows that… 3 billion + 2 billion = 5 billion …but doesn’t know that… 3,000,000,000 + 2,000,000,000 = 5,000,000,000 …has a “ language problem ”, not a “ math problem ”1 .

1 As a non-mathematical version of this, suppose there is a black cat in the room and you ask a person, “True or false: The cat is black?” Suppose the person answers “True”. Then you ask the same person “True or false: El gato es negro?” (Note: “El gato es negro” is Spanish for “The cat is black”). It turns out that the person doesn't understand Spanish so this time he answers “I don't know.” Now, if the person were blind, that would also explain why the answer was “I don’t know”. However, being blind doesn’t depend on language. Thus, if a person says “True” to the first question but “I don't know” to the second question the person has a language problem, not a © 2010 Herb I. Gross vision problem .

To help you further internalize the adjective / noun theme, consider the following question…

Have you ever seen a number?

What may seem amazing to you is that no one has ever seen a number.

Restated, let’s just say that we see numbers as

adjectives

rather than as nouns . That is, we have seen three apples, three dollars, three people, even three tally

marks

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Thus, the

theme of this presentation

begins with the underlying principle that numbers are adjectives that modify nouns (or other adjectives )

As an application of how we can use the “ numbers as adjectives ” principle, consider the following situation. Many students often fail to grasp the relative size of a

billion

dollars with respect to a

million

dollars in the sense that they view both amounts as being “

very big

”.

However, suppose we now let the adjectives “ million ” and “ billion ” both modify “ seconds ”. Some very elementary calculations show us that a million seconds is a “little less” than 12 days while a billion seconds “little more” than 31 years. This is a gives new life to the idea that although a million seconds is quite “big”, it is still a small fractional part of a billion seconds .

It also gives students (and others as well) an easy to understand ratio; namely, a million is to a billion as 12 days is to 31 years.

While it may be easy to confuse a million with a billion; no one ever confuses

12 days with 31 years!

However, size is relative. A quantity that is small relative to one quantity might be very large relative to another quantity.

Practice Problem #2

Using 400 days in a year as an estimate, how old would you have been when you had lived 1 million days?

Solution to Practice Problem #2 The answer is found by dividing 1 million (days) by 400 (our estimated number of days in a year). 1,000,000 ÷ 400 = 2,500 In other words, if there had been 400 days in a year, you would have been 2,500 years old when you had lived for 1 million days.

Notes on Practice Problem #2 The fact that there are less than 400 days in a year means that our estimate is less than the correct answer. Thus, without having to do any further computation, we are safe in saying that you will not have lived a million days until you were more than 2,500 years old.

Notes on Practice Problem #2 Taking into account leap years, there are 365 ¼ days in a year. With a calculator it takes no longer to divide by 365.25 than to divide by 400. Doing this, we would find that to the nearest whole number of years… 1,000,000 ÷ 365.25 = 2,738 © 2010 Herb I. Gross

A Note on Using a Calculator The calculator is a great aid in helping us to perform cumbersome computations quickly. It is not a replacement for logic and number sense. The user has to have enough knowledge to be able to tell the calculator what to compute. For example , the calculator will not divide 1,000,000 by 365.25 unless it is told to do so!

A Note on Using a Calculator Moreover, even with a calculator we can enter data incorrectly. To guard against this, it helps to have a number sense .

For example , in this discussion, our number sense told us that the millionth day since your birth cannot occur prior to the year 2500. Hence, the answer we got using a calculator (2738) is at least reasonable.

Notes on Practice Problem #2 The point we wish to make here is that to know the size of a quantity, we must know both the adjective and the noun it modifies. For example , even though a billion is more than a million, the fact remains that 1 billion seconds is less (approximately 31 years) time than 1 million days (approximately 2,700 years) © 2010 Herb I. Gross

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Key Point

In an era when education perhaps seems to be overly concerned with the use of manipulatives and high-tech visual aids, there is a tendency to forget that innovative use of language may in itself be both the greatest manipulative and the best visual aid that can ever exist; and of at least equal importance, it is available to be used by everyone.

In this course, we shall discuss how by treating mathematics as a language we can “translate” many complicated mathematical concepts into simpler but equivalent mathematical concepts. More specifically, we shall demonstrate how by properly choosing the nouns that numbers modify, we can make many mathematical concepts easier for students to comprehend, no matter what other modes of instruction we may be using.

Our previous discussion seems to demonstrate that the human mind A Note in Passing has the ability to comprehend concepts that cannot be seen. This ability is not limited to the study of numbers. For example , with today’s technology, we can measure time to the nearest billionth of a second, yet none of us has ever seen time!

What we have seen are the changes that occur with the passing of time.

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With our adjective /

noun

Mathematics as a Second Language

Transcript Mathematics as a Second Language

Arithmetic Revisited

Prelude to Mathematics as a Second Language

Note

Key Point

Have you ever seen a number?

Thus, the

theme of this presentation

begins with the underlying principle that numbers are adjectives that modify nouns (or other adjectives )

very big

Using 400 days in a year as an estimate, how old would you have been when you had lived 1 million days?

Key Point

With our adjective /

theme in mind, we will now begin our journey into the development of our present number system.

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